Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
1SchurParametrizationandOrthogonalModeling...
3
S=SP!n{H03H–13...}withtheinner-product(H–i3H–k)⌦=EH–i¯
H–k=hi,k=
hk,iwherehi,kdenotescovarianceoftherandomvariablesH–iandH–k.Consider
¯
thesetofrandomvariables{H03H–13...3H–n}andletH=[hi,k]i,k=0,...,ndenote
theGrammatrix,beingactuallyapositive-definiteHermitiancovariancematrixof
theprocessy.
Itwillprovetobeusefultonormalizethismatrixasfollowshi,kł
√⌘878⌘:7:
⌘87:
.
Thenweobtain
H=[hi,k]=
2
6
6
6
6
6
6
6
6
4
h1,01
hn,n...¯
¯
¯
1h0,1...
.
.
hn,n–1
...h
h0,n
n–1,n
1
.
.
.
3
7
7
7
7
7
7
7
7
5
stIt.
!⇠=[hk–i]=
2
6
6
6
6
6
6
6
4
hn¯
¯
h11...hn–1
¯
1
.
.
hn–1...1
h1...hn
.
.
.
.
.
3
7
7
7
7
7
7
7
5
...
...
.
.
...
.
(1.1)
TheToeplitzstructureofthestationaryprocesscovariancematriximpliesthe
followingclassificationofnon-stationaryprocesses,basedonanotionofdistance
betweenthecovariancematrixofagivenprocessandtheToeplitzmatrixofa
stationaryprocess,andintroducedemployingtheso-calledshift-matrix[2,10]
/=
2
6
6
6
6
6
6
6
4
00...0
10...0
0...10
.
.
.
.
.
3
7
7
7
7
7
7
7
5
.
...
...
Wethenhave
(1.2)
/H/t=
2
6
6
6
6
6
6
6
6
6
4
0
0
0¯
0¯
.
.
hn–1,0¯
h1,0
0
1
.
.
hn–1,1...
h0,1...h0,n–1
0
1
.
.
...
...h1,n–1
0
1
.
.
.
3
7
7
7
7
7
7
7
7
7
5
(1.3)
.
.
.
...
ConsidernowtheToeplitzmatrix⇠andcomputethefollowingdifierence-matrix
DH
=H–/H/t=
∆
2
6
6
6
6
6
6
6
4
hn0...0
h10...0
¯
¯
1h1...hn
.
.
.
.
.
.
.
3
7
7
7
7
7
7
7
5
.
.
...
(1.4)
NoticethattherankofthematrixDH(1.4)willbeequaltor!n:{DH}=2fora
wide-sensestationaryprocesswhilethismatrixitselfcanbeexpressedas
DH=[0
1¯
h1...¯
h1...¯
¯
hn]
hn
t[10
0–1][1h1...hn
0h1...hn]
(1.5)
